Geodesic Discrete Global Grid Systems Kevin Sahr, Denis White, and A. Jon Kimerling
ABSTRACT: In recent years, a number of data structures for global geo-referenced data sets have been proposed based on regular, multi-resolution partitions of polyhedra. We present a survey of the most promising of such systems, which we call Geodesic Discrete Global Grid Systems (Geodesic DGGSs). We show that Geodesic DGGS alternatives can be constructed by specifying five substantially indepen- dent design choices: a base regular polyhedron, a fixed orientation of the base regular polyhedron relative to the Earth, a hierarchical spatial partitioning method defined symmetrically on a face (or set of faces) of the base regular polyhedron, a method for transforming that planar partition to the corresponding spherical/ellipsoidal surface, and a method for assigning point representations to grid cells. The majority of systems surveyed are based on the icosahedron, use an aperture 4 triangle or hexagon partition, and are either created directly on the surface of the sphere or by using an equal- area transformation. An examination of the design choice options leads us to the construction of the Icosahedral Snyder Equal Area aperture 3 Hexagon (ISEA3H) Geodesic DGGS.
KEYWORDS: Discrete global grid systems, spatial data structures, global data models
Discrete Global Grid Systems: trarily regular DGGs by generating Voronoi cells on the surface of an ellipsoid from a specified set of Basic Definitions points. But, for many applications, it is desirable to have cells consisting of highly regular regions with Discrete Global Grid evenly distributed points. Regular DGGs are unbiased with respect to spatial Discrete Global Grid (DGG) consists of a patterns created by natural and human processes and set of regions that form a partition of the allow for the development of simple and efficient Earth’s surface, where each region has a algorithms. A single regular DGG may play multiple Asingle point contained in the region associated data structure roles. It may function as a raster data with it. Each region/point combination is a cell. structure, where each cell region constitutes a pixel. Depending on the application, data objects or It may serve as a vector data structure, where the vectors of values may be associated with regions, set of DGG points replaces traditional coordinate points, or cells. If an application defines only pairs (Dutton 1999). Each data object may be associ- the regions, the centroids of the regions form a ated with the smallest cell region in which it is fully suitable set of associated points. Conversely, if an contained, and these minimum bounding cells may application defines only the points, the Voronoi then be used as a coarse filter in operations such regions of those points form an obvious set of asso- as object intersection. The DGG can also be used ciated cell regions. as a useful graph data structure by taking the DGG Applications often use DGGs with cell regions that points as the graph vertices and then connecting are irregular in shape and/or size. For example, the points associated with neighboring cells with unit- division of the Earth’s surface into land masses and weight edges. bodies of water constitutes one of the most important The most commonly used regular DGGs are those DGGs. A more general example, the Hipparchus based on the geographic (latitude–longitude) coor- System (Lukatella 2002), allows the creation of arbi- dinate system. Raster global data sets often employ cell regions with edges defined by arcs of equal-angle Kevin Sahr is an Assistant Professor in the Department of Computer increments of latitude and longitude (for example, Science at Southern Oregon University, Ashland, OR 97520. E-mail: the 2.50 x 2.50, 50 x 50, and 100 x 100 NASA Earth
Cartography and Geographic Information Science, Vol. 30, No. 2, 2003, pp. 121-134 (Hastings and Dunbar 1998)). Similarly, vector data sets that employ a geographic coordinate system to define location values commonly choose a specific precision for those values. The choice of a specific precision for geographic coordinates forms an implicit grid of fixed points at regular angular increments of latitude and longitude, and a particular data set of that precision can consist only of coordinate values chosen from this set of fixed points. Ideally, the regions associated with the geographic vector coordinate points would be the corresponding Voronoi regions on the Earth’s surface, although they are more commonly the corresponding Voronoi regions on a surrogate representation of the Earth’s surface, such as a sphere or ellipsoid. In practice, applications often implicitly employ Voronoi regions defined on the longitude x latitude plane, on which the regions are, conveniently, regular planar squares. As we shall see, employing a surrogate representation for the Earth’s surface on which the cell regions are Figure 1. A portion of two resolutions (100 and 10 precision) of regular planar polygons is a useful and common the DGGS implicitly generated by multiple precisions of decimal approach in DGG construction. geographic coordinate system vector representations. Note that this is an incongruent, aligned hierarchy. Discrete Global Grid System A Discrete Global Grid System (DGGS) is a Discrete Global Grid Systems based on the geo- series of discrete global grids. Usually, this series graphic coordinate system have numerous practical consists of increasingly finer resolution grids; advantages. The geographic coordinate system has i.e., the grids in the series have a monotonically been used extensively since well before the computer increasing number of cells. If the grids are defined era and is therefore the basis for a wide array of consistently using regular planar polygons on a existing data sets, processing algorithms, and soft- surrogate surface, we can define the aperture of a ware. Grids based on square partitions are by far the DGGS as the ratio of the areas of a planar polygon most familiar to users, and they map efficiently to cell at resolution k and at resolution k+1 (this is common data structures and display devices. a generalization of the definition given in Bell et But such grids also have limitations. Discrete al.(1983)). Later we will discuss DGGSs that have Global Grid Systems induced by the latitude–lon- more than one type of polygonal cell region. In gitude graticule do not have equal-area cell regions, these cases there is always one cell type that clearly which complicates statistical analysis on these grids. predominates, and the aperture of the system is The cells become increasingly distorted in area, defined using the dominant cell type. shape, and inter-point spacing as one moves north Kimerling et al. (1999) and Clarke (2002) note and south from the equator. The north and south the importance of regular hierarchical relationships poles, both points on the surface of the globe, map between DGGS resolutions in creating efficient data to lines on the longitude x latitude plane; the top structures. Two types of hierarchical relationship and bottom row of grid cells are, in fact, triangles, are common. A DGGS is congruent if and only if not squares as they appear on the plane. These polar each resolution k cell region consists of a union of singularities have forced applications such as global resolution k+1 cell regions. A DGGS is aligned if climate modeling to make use of special grids for and only if each resolution k cell point is also a cell the polar regions. Square grids in general do not point in resolution k+1. If a DGGS does not have exhibit uniform adjacency; that is, each square grid these properties, the system is defined as incongru- cell has four neighbors with which it shares an edge ent or unaligned. For example, the most widely used and whose centers are equidistant from its center. DGGS is generated implicitly by multiple precisions Each cell, however, also has four neighbors with of decimal geographic vector representations. This which it shares only a vertex and whose centers are DGGS has an aperture of 10 and is incongruent and a different distance from its center than the distance aligned (Figure 1). to the centers of the edge neighbors. This compli-
122 Cartography and Geographic Information Science Vol. 30, No. 2 123 store raster data sets, but they may also be used as a substitute for traditional coordinate-based vector data structures (Dutton 1999), as data containers, or as the basis for graphs (as described in the previous section). Geodesic DGGSs have been used to develop Figure 2. Planar and spherical versions of the five platonic solids: the tetrahedron, statistically sound survey hexahedron (cube), octahedron, dodecahedron, and icosahedron. sampling designs on the Earth’s surface (Olsen et al. 1998), for optimum path cates the use of these grids for such applications as determination (Stefanakis discrete simulations. and Kavouras 1995), for line simplification (Dutton Attempts have been made to create DGGs based 1999), for indexing geospatial databases (Otoo and on the geographic coordinate system but adjusted Zhu 1993; Dutton 1999; Alborzi and Samet 2000), to address some of these difficulties. For example, and for the generation of spherical Voronoi diagrams Kurihara (1965) decreased the number of cells with (Chen et al. 2003). They have also been proposed increasing latitude so as to achieve more consistent as the basis for dynamic simulations such as those cell region sizes. Bailey (1956), Paul (1973), and used in global climate modeling (Williamson 1968; Brooks (1981) used similar adjustments of latitude Sadournay et al. 1968; Heikes and Randall 1995a, and/or longitude cell edges to achieve cell regions 1995b; Thuburn 1997). with approximately equal areas. But these schemes It is highly unlikely that any single Geodesic DGGS achieved more regular cell region areas at the cost of will ever prove optimal for all applications. Many more irregular cell region shapes and more complex of the proposed systems include design innovations cell adjacencies. Tobler and Chen (1986) projected in particular areas, though their construction may the Earth onto a rectangle using a Lambert cylin- have involved other, less desirable design choices. drical equal area projection and then recursively Therefore, rather than surveying individual Geodesic subdivided that rectangle, but, as in the case of the DGGSs as monolithic, closed systems, we will take other mentioned approaches, this did not address the approach here of viewing the construction of a the basic problem that the sphere/ellipsoid and the Geodesic DGGS as a series of design choices which plane are not topologically equivalent. are, for the most part, independent. The following five design choices fully specify a Geodesic DGGS: 1. A base regular polyhedron; Geodesic Discrete Global 2. A fixed orientation of the base regular Grid Systems polyhedron relative to the Earth; 3. A hierarchical spatial partitioning method The inadequacies of DGGSs based on the geo- defined symmetrically on a face (or set of graphic coordinate system have led a number faces) of the base regular polyhedron; of researchers to explore alternative approaches. 4. A method for transforming that planar Many of these approaches involve the use of partition to the corresponding spherical/ regular polyhedra as topologically equivalent sur- ellipsoidal surface; and rogates for the Earth’s surface, and, in our opinion, 5. A method for assigning points to grid cells. these attempts have led to the most promising We will now look at each of these design choices in known options for DGGSs. A number of research- turn, discussing the decisions made in the develop- ers have been inspired directly or indirectly by ment of a number of Geodesic DGGSs. R. Buckminster Fuller’s work in discretizing the sphere, which led to his development of the geo- Base Regular Polyhedron desic dome (Fuller 1975). We will thus refer to this class of DGGSs as Geodesic Discrete Global Grid As White et al. (1992) and many others have Systems. observed, the spherical versions of the five platonic Geodesic DGGSs have been proposed for a number solids (Figure 2) represent the only ways in which of specific applications. Inherently regular in design, the sphere can be partitioned into cells, each con- these systems have most commonly been used to sisting of the same regular spherical polygon, with
122 Cartography and Geographic Information Science Vol. 30, No. 2 123 the same number of polygons meeting at each vertex. The platonic solids have thus been commonly used to construct Geodesic DGGSs, although other regular polyhedra have some- times been employed. Among these the truncated icosahedron has proved to be popular (White et al. 1992). It should be noted, however, that an equivalently partitioned DGGS could be constructed using the icosahe- dron itself. The other regular polyhedra remain unexplored Figure 3a. Spherical icosahedron (in Mollweide projection) oriented with vertices for DGGS construction, so we at poles and an edge aligned with the prime meridian. Note the lack of symmetry limit our discussion here to the about the equator. platonic solids. In general, platonic solids with smaller faces reduce the distortion introduced when transforming between a face of the polyhedron and the corresponding spherical surface (White et al. 1998). The tetrahedron and cube have the larg- est face size and are thus relatively poor base approximations for the sphere. But because the faces of the cube can be easily subdivided into square quadtrees, it was chosen as the base platonic solid by Alborzi and Samet (2000). The icosahedron has Figure 3b. Spherical icosahedron (in Mollweide projection) oriented using Fuller’s the smallest face size and, therefore, Dymaxion orientation. Note that all vertices fall in the ocean. any DGGSs defined on it tend to display relatively small distortions. The icosahedron is thus the most common choice for a base platonic solid. Geodesic DGGSs based on the icosahedron include those of Williamson (1968), Sadournay et al. (1968), Baumgardner and Frederickson (1985), Sahr and White (1998), White et al. (1998), Fekete and Treinish (1990), Thuburn (1997), White (2000), Song et al. (2002), and (with an adjustment as discussed in the next section) Heikes and Randall (1995a, 1995b). Dutton chose the octahedron as Figure 3c. Spherical icosahedron (in Mollweide projection) oriented for symmetry the base polyhedron for the Global about equator by placing poles at edge midpoints. Note the symmetry about the Elevation Model (1984) and for the equator and the single vertex falling on land. Quaternary Triangular Mesh (QTM) system (1999), while Goodchild and hedron has the advantage that it can be oriented Yang (1992) based a similar system on it, and White with vertices at the north and south poles, and at the (2000) used it as an alternative base solid. The octa- intersection of the prime meridian and the equator,
124 Cartography and Geographic Information Science Vol. 30, No. 2 125 aligning its eight faces with the spherical octants presumably due to the asymmetry in the underly- formed by the equator and prime meridian. Given a ing icosahedron. To counter this effect they rotate point in geographic coordinates, it is then trivial to the southern hemisphere of the icosahedron by 36 determine on which octahedron face the point lies, degrees, and the resulting “twisted icosahedron” is but, because the octahedron has larger faces than symmetrical about the equator. the icosahedron, projections defined on the faces Fuller (1975) chose an icosahedron orientation of the octahedron tend to have higher distortion (Figure 3b) for his Dymaxion icosahedral map (White et al. 1998). projection that places all 12 of the icosahedron Wickman et al. (1974) observe that if a point is placed vertices in the ocean so that the icosahedron can in the center of each of the faces of a dodecahedron be unfolded onto the plane without ruptures in and then raised perpendicularly out to the surface any landmass. This is the only known icosahedron of a circumscribed sphere (“stellated”), each of the orientation with this property. Note that one com- 12 pentagonal faces becomes 5 isosceles triangles. pact way of specifying the orientation of a platonic The stellated dodecahedron thus has 60 triangular solid is by giving the geographic coordinates of faces compared to the 20 faces of the icosahedron, one of the polyhedron’s vertices and the azimuth and an equal area projection can be defined on the from that vertex to an adjacent vertex. For platonic smaller faces of the stellated dodecahedron with solids this information will completely specify the lower distortion than on the icosahedron (e.g., Snyder position of all the other vertices. Using this form 1992). However, the triangular faces are no longer of specification, Fuller’s Dymaxion orientation can equilateral and therefore such a projection displays be constructed by placing one vertex at 5.24540W inconsistencies along the edges between faces. longitude, 2.30090N latitude and an adjacent one at an azimuth of 7.466580 from the first vertex. Polyhedron Orientation We note that if the icosahedron is oriented so that the north and south poles lie on the midpoints of Once a base polyhedron is chosen, a fixed orienta- edges rather than at vertices, then it is symmetrical tion relative to the actual surface of the Earth must about the equator without further adjustment. While be specified. Alborzi and Samet (2000) oriented maintaining this property we can minimize the number the cube by placing face centers at the north and of icosahedron vertices that fall on land, following south poles. White et al. (1992) oriented the trun- Fuller’s lead. The minimal case appears to be an cated icosahedron such that a hexagonal face cov- orientation (Figure 3c) that has only one vertex on ered the continental United States. Dutton (1984, land, in China’s Sichuan Province. This orientation 1999), and Goodchild and Yang (1992) oriented can be constructed by placing one vertex at 11.250E the octahedron so that its faces align with the longitude, 58.282520N latitude and an adjacent one octants formed by the equator and prime merid- at an azimuth of 0.00 from the first vertex. ian. Wickman et al. (1974) oriented the dodecahe- dron by placing the center of a face at the north pole and a vertex of that face on the prime merid- Spatial Partitioning Method ian, thus aligning with the prime meridian an Once we have a base regular polyhedron, we must edge of one of the triangles created by stellating next choose a method of subdividing this polyhe- the dodecahedron. dron to create multiple resolution discrete grids. In the case of the icosahedron, the most common In the case of platonic solids one can define the orientation (Figure 3a) is to place a vertex at each of subdivision methodology on a single face of the the poles and then align one of the edges emanating polyhedron or on a set of faces that constitute a from the vertex at the north pole with the prime unit that tiles the polyhedron, provided that the meridian. This orientation is used by Williamson subdivision is symmetrical with respect to the face (1968), Sadournay et al. (1968), Fekete and Treinish or tiling unit. Four partition topologies have been (1990), and Thuburn (1997). used: squares, triangles, diamonds, and hexagons. Heikes and Randall’s (1995a,b) icosahedron-based Alborzi and Samet (2000) performed an aperture system was developed specifically for performing 4 subdivision to create a traditional square quadtree global climate change simulations. They note that in on each of the square faces of the cube. We have the common vertices-at-poles icosahedron placement observed that the preferred choices for base platonic (Figure 3a) the icosahedron is not symmetrical about solid[s] are the icosahedron, the octahedron, and the equator. When a simulation on a DGGS with the stellated dodecahedron, each of which has a this orientation is initialized to a state symmetrical triangular face. The obvious choice for a triangle is about the equator, and then allowed to run, it evolves to subdivide it into smaller triangles. Like the square, into a state that is asymmetrical about the equator, an equilateral triangle can be divided into n2 (for
124 Cartography and Geographic Information Science Vol. 30, No. 2 125 any positive integer n) smaller equilateral tri- angles by breaking each edge into n pieces and connecting the break points with lines par- allel to the triangle edges (Figure 4). In geodesic dome litera- ture this is referred to as a Class I or alternate subdivision (Kenner Figure 4. Three levels of a Class I aperture 4 triangle hierarchy defined on a single triangle 1976). Recursively sub- face. dividing the triangles thus obtained gener- ates a congruent and aligned DGGS with aperture n2. Small apertures have the advantage of generat- ing more grid resolutions, thus giving applications more resolutions from which to choose. For congruent triangle subdivision the smallest possible aperture is 4 (n Figure 5. Three levels of a Class II aperture 3 triangle hierarchy defined on a single triangle = 2). This aperture also face. conveniently parallels the fourfold recursive subdivi- thus a foreign alternative for many potential users, sion of the square grid quadtree; many of the algorithms and they do not display as efficiently as squares on developed on the square grid quadtree are transferable common output display devices that are based on to the triangle grid quadtree with only minor modifica- square lattices of pixels. Like square grids, they do tions (Fekete and Treinish 1990; Dutton 1999). This subdivision approach (Figure 4) was used by Wickman not exhibit uniform adjacency, each cell having three et al. (1974), Baumgardner and Frederickson (1985), edge and nine vertex neighbors. Unlike squares, the Goodchild and Yang (1992), Dutton (1999), Fekete cells of triangle-based discrete grids do not have and Treinish (1990), White et al. (1998), and Song et uniform orientation; as can be seen in Figures 4 and al. (2002). Congruent and unaligned Class I aperture 5, some triangles point up while others point down, 9 (n = 3) triangle hierarchies have been proposed by and many algorithms defined on triangle grids must White et al. (1998) and Song et al. (2002). take into account triangle orientation. An aperture 3 triangle subdivision is also possible. While the square is the most popular cell region In this approach, referred to as the Class II or tria- shape for planar discrete grids, its geometry makes con subdivision (Kenner 1976), each triangle edge it unusable on the triangle-faced regular polyhedra is broken into n = 2m pieces (where m is a positive that we have seen are preferred for constructing integer). Lines are then drawn perpendicular to Geodesic DGGSs. However, White (2000) notes that the triangle edges to form the new triangle grid pairs of adjacent triangle faces may be combined to (Figure 5). The Class II breakdown is incongruent form a diamond or rhombus, and this diamond may and unaligned. No Geodesic DGGSs have been be recursively sub-divided in a fashion analogous to proposed based on this partition, though the verti- the square quadtree subdivision (Figure 6). When ces of a Class II breakdown have been used as cell one begins with either the octahedron or icosahe- points by Dutton (1984) and by Williamson (1968) dron, this yields a congruent, unaligned Geodesic to construct a dual hexagon grid. DGGS with aperture 4. Because diamond-based grids Triangles have a number of disadvantages as the have a topology identical to square-based quadtree basis for a DGGS. First, they are not squares; they are grids they can take direct advantage of the wealth
126 Cartography and Geographic Information Science Vol. 30, No. 2 127 polygon will be formed at each of the polyhedron’s vertices. The number of such polygons, corre- sponding to the number of polyhedron vertices, will remain constant regard- less of grid resolution. In the case of an octahedron these polygons will be eight squares, in the case Figure 6. Three levels of an aperture 4 diamond hierarchy. The coarsest diamond resolution of the icosahedron they consists of two triangle faces as indicated. will be 12 pentagons. While single-resolution, of quadtree-based algorithms. But like square grids, hexagon-based discrete they do not display uniform adjacency. grids are becoming increasingly popular, the use of The hexagon has received a great deal of recent multi-resolution, hexagon-based discrete grid sys- interest as a basis for planar discrete grids. Among tems has been hampered by the fact that congruent the three regular polygons that tile the plane (tri- discrete grid systems cannot be built using hexagons; angles, squares, and hexagons), hexagons are the most it is impossible to exactly decompose a hexagon compact, they quantize the plane with the smallest into smaller hexagons (or, conversely, to aggregate average error (Conway and Sloane 1988), and they small hexagons to form a larger one). Hexagons can provide the greatest angular resolution (Golay 1969). be aggregated in groups of seven to form coarser- Unlike square and triangle grids, hexagon grids do resolution objects which are almost hexagons (Figure have uniform adjacency; each hexagon cell has six 7), and these can again be aggregated into pseudo- neighbors, all of which share an edge with it, and hexagons of even coarser resolution, and so on. all of which have centers exactly the same distance Known as Generalized Balanced Ternary (Gibson away from its center. Each hexagon cell has no and Lucas 1982), this structure has become the most neighbors with which it shares only a vertex. This widely used planar multi-resolution, hexagon-based fact alone has made hexagons increasingly popular grid system. However, it has several problems as a as bases for discrete spatial simulations. Frisch et al. general-purpose basis for spatial data structures. The (1986) argue that the six discrete velocity vectors of first is that the cells are hexagons only at the finest the hexagonal lattice are necessary and sufficient to resolution. Secondly, the finest resolution grid must simulate continuous, isotropic, fluid flow. A recent be determined prior to creating the system, and once textbook (Rothman and Zaleski 1997) on fluid flow determined it is impossible to extend the system cellular automata is based entirely on hexagonal to finer resolution grids. Thirdly, the orientation meshes, with discussions of square meshes included of the tessellation rotates by about 19 degrees at “only for pedagogical calculations.” Triangle grids, each level of resolution. Finally, it does not appear which are even more insufficient for this purpose, to be possible to symmetrically tile triangular faces are not mentioned. with such a hierarchy, which makes it unusable as a Studies by GIS researchers (Kimerling et al. 1999) subdivision choice for a Geodesic DGGS. and mathematicians (Saff and Kuijlaars 1997) indi- There are, however, an infinite series of apertures cate that many of the advantages of planar hexagon that produce regular hierarchies of incongruent, grids may carry over into hexagon-based Geodesic aligned hexagon discrete grids. It should be noted DGGSs. A hexagon-based grid has been adopted by that, as these hierarchies are incongruent, they do not the U.S. EPA for global sampling problems (White et naturally induce hierarchical data structures which al. 1992). And hexagon-based Geodesic DGGSs have are trees, and thus common tree-based algorithms been proposed at least four times in the atmospheric cannot be directly adapted for use on these hexagon modeling literature (Williamson 1968; Sadournay et hierarchies. But it should also be noted that, as indi- al. 1968; Heikes and Randall 1995a, 1995b; Thuburn cated previously, traditional multi-resolution vector 1997)—to our knowledge more often than any other data structures such as the geographic coordinate Geodesic DGGS topology. It should be noted that system are also incongruent and aligned. This may it is impossible to completely tile a sphere with indicate that hexagon grids are more appropriate hexagons. When a base polyhedron is tiled with for vector applications than congruent, unaligned hexagon-subdivided triangle faces, a non-hexagon triangle and diamond hierarchies.
126 Cartography and Geographic Information Science Vol. 30, No. 2 127 Aperture 4 is the most common choice for hexagon-based DGGSs. Figures 8 and 9 illustrate aperture 4 hexagon subdivisions corresponding to the Class I and Class II symmetry axes, respectively. The DGGSs of Heikes and Randall (1995a) and Thuburn (1997) are Class I aperture 4 hexagon grids, while Williamson (1968) uses a Class II aperture 4 hexagon grid. As noted above, small apertures have the advantage of allowing more potential grid choices. Aperture 3 is the smallest aperture that yields an aligned hexagon hierarchy (Figure 10). In aperture 3 hierarchies the orienta- tion of hexagon grids alternates in successive resolutions between Class I and Class II. Aperture 3 hexagon Geodesic DGGS have been proposed by a number of researchers, including Sahr and White (1998). White et al. (1992) proposed hexagon grids of aperture 3, 4, or 7, and White et al. (1998) discussed hexagon grids of aperture 4 (Class I) and 9 (Class II). Figure 7. Seven-fold hexagon aggregation into coarser pseudo-hexagons. Sadournay et al. (1968) used a Class I hexagon grid of arbitrary aperture, which is incongruent and unaligned. Perhaps the simplest approach is to perform the We refer to this approach as an n-frequency hierar- desired partition directly on the spherical surface, chy. Note that it is possible to construct incongruent, using great circle arcs corresponding to the cell unaligned n-frequency hierarchies using triangles edges on the planar face(s). The aperture 4 Class and diamonds as well, though, to our knowledge, I triangle subdivision can be performed on the this has not been proposed. sphere by connecting the midpoints of the edges Figure 11 illustrates the most common partitioning of the base spherical triangle and then, recursively methods defined on an icosahedron and projected performing the same operation on each of the to the sphere using the inverse Icosahedral Snyder resulting triangles. This technique was used by Baumgardner and Frederickson (1985) and Fekete Equal Area (ISEA) Projection (Snyder 1992). and Treinish (1990). Dutton (1984) performed a Class II triangle subdivision on the surface of the Transformation octahedron and then adjusted the vertices to reflect Once a partitioning method has been specified on the point elevations. a face or faces of the base polyhedron, a transfor- While this straightforward approach works for mation must be chosen for creating a similar topol- creating an aperture 4 triangle subdivision, it is ogy on the corresponding spherical or ellipsoidal important to note that, in general, sets of great surface. There are two basic types of approaches circle arcs corresponding to the edges of planar (Kimerling et al. 1999). Direct spherical subdivision triangle partitions do not intersect in points on the approaches involve creating a partition directly surface of the sphere, as they do on the plane. More on the spherical/ellipsoidal surface that maps to complicated methods are needed to form spherical the corresponding partition on the planar face(s). partitions analogous to some of the other planar Map projection approaches use an inverse map partitions we have discussed. projection to transform a partition defined on the Williamson (1968) used great circle arcs corre- planar face(s) to the sphere/ellipsoid. White et al. sponding to two of the three sets of Class II triangle (1998) provide a comparison of the area and shape subdivision grid lines to determine a set of triangle distortion that occurs under a number of different vertices and then formed the last set of grid lines transformation choices. by connecting the existing vertices with great circle
128 Cartography and Geographic Information Science Vol. 30, No. 2 129 Figure 8. Three levels of a Class I aperture 4 hexagon hierarchy defined on a single triangle face.
Figure 9. Three levels of a Class II aperture 4 hexagon hierarchy defined on a single triangle face.
arcs. These triangle vertices form the center points the midpoints of the base spherical triangle to form of the dual Class II aperture 4 hexagon grid (the cell the first resolution of a Class I aperture 4 triangle edges of which are not explicitly defined). grid. They then broke each of the new edges at the Sadournay et al. (1968) created an aperture m midpoint into two great circle arcs and adjusted the (where m = n2 for some positive integer n) Class I position of the breakpoint to achieve equal area quasi- triangle subdivision on the sphere by breaking each triangles. This procedure is then applied recursively edge of the base spherical triangle into n segments to yield an equal-area DGGS. Rather than using and connecting the breakpoints of two of the edges great circle arcs for triangle subdivision, Song et al. with great circle arcs. These arcs are then subdivided (2002) proposed using small circle arcs optimized evenly into segments corresponding to the planar to achieve equal cell region areas. subdivision. The resulting breakpoints form the Heikes and Randall (1995a) constructed a Class centers of the dual Class I hexagon grid. Thuburn I aperture 4 hexagon grid by taking the spherical (1997) performed a Class I aperture 4 triangle sub- Voronoi of the vertices of a Class I aperture 4 triangle division and then calculated the spherical Voronoi subdivision on their twisted icosahedron. They then cells of the triangle vertices to define the dual Class adjusted the grid using an optimization scheme to I aperture 4 hexagon grid. improve its finite difference properties for use in A number of researchers have attempted to global climate modeling. adjust the grids created using great circle arcs to White et al. (1998) evaluated a number of methods meet application-specific criteria. For instance, for for constructing triangle subdivisions on spherical many applications it would be desirable for the cell triangles and observed that using appropriate inverse regions of each discrete grid resolution to be equal map projections to transform a subdivided planar in area; the grids discussed above do not have this triangle onto a spherical triangle may be more effi- property. Wickman et al. (1974) began by connecting cient than using recursively defined procedures. Any
128 Cartography and Geographic Information Science Vol. 30, No. 2 129 Figure 10. Three levels of an aperture 3 hexagon hierarchy defined on a single triangle face. Note the alternation of hexagon orientation (Class II, Class I, Class II, etc.) with successive resolutions.
projection may be used, provided that it maps the corresponding great-circle arc edges and, therefore, straight-line planar face edges to the great-circle arc does not create a true Geodesic DGGS. edges of the corresponding spherical face. There are at least four projections with this prop- Assigning Points to Grid Cells erty. The common gnomonic projection has this property for all polyhedra but exhibits relatively When specified, the points associated with grid large area and shape distortion. Snyder (1992) cells are usually chosen to be the center points developed equal area projections defined on all of of the cell regions. If an inverse map projection approach is used to transform the cells from the the platonic solids, but with greater shape distor- planar faces to the sphere, then it is often conve- tion and more irregular spherical cell edges than nient to choose the center points of the planar cell the equal-area method of Song et al. (2002). On regions (which do not, in general, correspond to the icosahedron, the implementation of Fuller’s the cell region centroids on the Earth’s surface) Dymaxion map projection (Fuller 1975) given in so that the points form a regular lattice, at least Gray (1995) also has the required property but with on patches of the plane. If the cells are formed less area and shape distortion than the gnomonic by direct spherical subdivision, the choice of projection and less shape distortion than Snyder’s points may be complicated by the counter-intui- icosahedral projection, though the Fuller/Gray tive behavior of great-circle arcs described above. projection is not equal area. Goodchild and Yang Gregory (1999) discussed several alternatives (1992) used a Plate Carree projection to project the for point selection in the case of direct spherical faces of the octahedron to the sphere, and Dutton subdivision. Dutton’s (1984) GEM DGGS used (1999) developed the Zenithial OrthoTriangular points that are the vertices of a Class II triangle (ZOT) projection for the same purpose. subdivision. As described in the previous sub-sec- White et al. (1998) constructed Class I aperture 4 tion, the hexagonal DGGSs of Williamson (1968), and 9 triangle grids on planar icosahedral faces. They Sadournay et al. (1968), Heikes and Randall also constructed a Class I aperture 4 hexagon grid by (1995a), and Thuburn (1997) all specify cell center taking the dual of the aperture 4 triangle grid and a points as the vertices of a dual spherical triangle Class II aperture 9 hexagon grid by aggregating the grid. The hexagonal grid cell boundaries, when cells of the aperture 9 triangle grid. In all cases they specified, are created by calculating the associated transformed the resulting cells to the sphere, using spherical Voronoi cells. direct spherical subdivision or the inverse gnomonic, Fuller/Gray, or Icosahedral Snyder Equal Area (ISEA) map projections. Summary and Conclusions White et al. (1992) and Alborzi and Samet (2000) used the inverse Lambert Azimuthal Equal Area Table 1 summarizes the design choices that define projection to project the faces of the truncated each of the Geodesic DGGSs we have discussed. icosahedron and cube to the sphere. White et al. Note that the number of options employed to con- (1992) noted, however, that this projection does struct a Geodesic DGGS is actually rather small, not map the straight-line planar face edges to the and certain choices clearly predominate in the
130 Cartography and Geographic Information Science Vol. 30, No. 2 131 existing designs. In particular, the icosahedron is clearly the popu- lar choice for a base polyhedron; it is used in 10 of the 16 listed grid designs. Methods based on direct spherical subdivision are employed by about half of the Vertices
Not Specified Not Specified Not Specified Not Specified Not Specified Not Specified Not Specified Not Specified Not Specified Not Specified grid designs. Also popular are
Point Assignment Point equal area transformations, which Class II Triangle Vertices Class II Triangle are used by six of the grids. The Modified Class II Triangle Vertices Modified Class II Triangle -frequency Class I Triangle Vertices -frequency Class I Triangle Twisted Class I Aperture 4 Triangle Class I Aperture 4 Triangle Twisted Class I Aperture 4 Triangle Vertices Class I Aperture 4 Triangle grid designs are almost evenly n Cell Centers in ISEA Projection Space Cell Centers in ISEA Projection split between triangle and hexa- gon partitions, but the diamond partition is a recent design that may yet prove popular due to its direct relationship to the square quadtree. We have shown that a Geodesic ZOT ISEA DGGS can be specified through a Subdivision Subdivision Plate Carree Not Specified
Transformation very small number of design choices, each of which is relatively indepen- Optimized Direct Spherical Direct Spherical Subdivision Direct Spherical Subdivision Direct Spherical Subdivision Direct Spherical Subdivision Direct Spherical Subdivision Direct Spherical Subdivision Direct Spherical Subdivision, Lambert Azimuthal Equal Area Lambert Azimuthal Equal Area Area-adjusted Direct Spherical Gnomonic, ISEA, or Fuller/Gray dent of the others. In effect, future Equal Area Small Circle Subdivision Geodesic DGGS designers may pick- and-choose from the menu of design choices to construct a DGGS to meet their specific application needs. As an example, let us take each of the design decisions in turn and attempt to construct a good general-purpose (Class I) (Class I) Hexagon Partition Geodesic DGGS. Aperture 4 Square
Aperture 4 Triangle First, due to its lower distortion Aperture 3 Hexagon Aperture 9 Hexagon Aperture 4 Diamond Class I Aperture 4 or II Implied Class II Aperture 4 characteristics we choose the icosa-
-frequency Hexagon (Class I) Aperture 3, 4, or 7 Hexagon Aperture 4 Triangle (Class I) Aperture 4 Triangle (Class I) Aperture 4 Triangle (Class I) Aperture 4 Triangle (Class I) Aperture 4 Triangle Implied Aperture 3 Hexagon Twisted Aperture 4 Hexagon Twisted Aperture 4 Hexagon (Class I) n Class I Aperture 4 or 9 Triangle Class I Aperture 4 or 9 Triangle or hedron for our base platonic solid. We orient it with the north and south poles lying on edge midpoints, such that the resulting DGGS will be symmetrical about the equator. pole
CONUS Next we select a suitable partition. Unspecified Unspecified Orientation Not Specified Not Specified
Octant aligned Octant aligned Octant aligned The hexagon partition has numerous Vertices or Face or Face Vertices Centers at Poles Vertices at poles Vertices at poles Vertices at poles Vertices at poles Vertices Equator symmetric Stellation vertex on
Poles on face centers Poles advantages, and we choose aperture 3, Hexagon face covering the smallest possible aligned hexagon aperture. Because equal-area cells are advantageous for many applications, we choose the inverse ISEA projec- Cube Twisted Twisted Stellated
Truncated Truncated tion to transform the hexagon grid Octahedron Octahedron Octahedron Octahedron Icosahedron Icosahedron Icosahedron Icosahedron Icosahedron Icosahedron Icosahedron Icosahedron Icosahedron Icosahedron Dodecahedron Icosahedron or to the sphere, and we specify that Base Polyhedron each DGGS point lies at the center of the corresponding planar cell region. We call the resulting grid the ISEA Aperture 3 Hexagonal (ISEA3H) DGGS. Figure 12 shows
& 1995b the ETOPO5 global elevation data Reference White 2000 Dutton 1984 Dutton 1999 Thuburn 1997 Baumgardner & Song et al. 2002 set (Hastings and Dunbar 1998) Williamson 1968 White et al. 1992 White et al. 1998 Frederickson 1985 Frederickson Sahr & White 1998 Sadourny et al. 1968 Wickman et al. 1974 Alborzi & Samet 2000 Fekete & Treinish 1990 & Treinish Fekete Goodchild & Yang 1992 Goodchild & Yang
Heikes & Randall 1995a & Randall Heikes binned into four resolutions of the ISEA3H DGGS. The elevation value Table 1. Summary of Geodesic DGGS design choices.
130 Cartography and Geographic Information Science Vol. 30, No. 2 131 Figure 11. Three resolutions of icosahedron-based Geodesic DGGS’s using four partition methods: (a) Class I aperture 4 triangle, (b) aperture 4 diamond, (c) aperture 3 hexagon, and (d) Class I aperture 4 hexagon.
for each ISEA3H cell was calculated by taking the In particular, we feel that further research into arithmetic mean of all ETOPO5 data points that fall transformations for Geodesic DGGS definition is into that cell region. More information on this and required. For example, a DGGS projection that other ISEA-based Geodesic DGGSs may be found is equal area, but has less shape distortion than at http://www.sou.edu/cs/sahr/dgg. the ISEA projection, would be very desirable. Additionally, the grids discussed here are defined with reference to the sphere; many applications Directions for Further Research will require more accurate definitions referenced While such studies as White et al. (1998), Kimerling to ellipsoids. And as specific grids are chosen for et al. (1999), Clarke (2002), and the current work practical use efficient transformations must be have made significant steps in defining and evalu- defined that will allow data to be moved between ating existing DGGS alternatives there remain a grids while preserving data quality. number of areas that we believe require further Existing studies have treated DGGSs from the per- research. First, it should be noted that additional spective of the broader GIS community, but effective research may reveal new design choice alterna- evaluation of design alternatives can only take place tives that are superior to those already proposed. in the context of specific applications and end-user
132 Cartography and Geographic Information Science Vol. 30, No. 2 133 Figure 12. ETOPO5 5’ global elevation data binned into the ISEA3H Geodesic DGGS at four resolutions with approximate hexagon areas of: (a) 210,000 km2, (b) 70,000 km2, (c) 23,000 km2, and (d) 7,800 km2.
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